He heads back to the Scottish Premiership club for his second spell.
Rodgers initially left the club in 2019 to join Leicester, but was sacked by the Foxes amid a disastrous Premier League campaign that ultimately ended in relegation.
The 50-year-old has signed a three-year contract with Celtic.
Rodgers said: "I am really delighted to return to Celtic and I am hugely excited by this great opportunity. When I was given the privilege of being asked to join the club again, it was a very simple decision for me and my family.
"We have enjoyed some great times at Celtic before and this will be my goal again, to deliver good football, ensure we have a team we all love to watch and ultimately bring more success to our fans.
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"Ange has done some brilliant work across the last two years and I will be doing all I can to maintain Celtic's momentum as we face into all our domestic and European challenges.
"We have a hugely impressive and talented group of players and staff at the club, and I am really looking forward to meeting up with them all and then getting straight down to work.
"I wanted John to be with me as we move forward. He is a very talented coach, someone I know well and respect and he will be very important to myself and our players.
"The club is in fantastic condition at all levels and I would like to thank the board for putting their trust in me to take the team forward. We will work very closely together as we strive to bring that success to our supporters.
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"Let's get to work!"